Exponential Convergence to the Maxwell Distribution For Some Class of Boltzmann Equations

Details Exponential Convergence to the Maxwell Distribution For Some Class of Boltzmann Equations Exponential Convergence to the Maxwell Distribution For Some Class of Boltzmann Equations

2010-11-18

arxiv.org/abs/1011.4194v1

Mathematics

Analysis of PDEs

36 Pages, 2 figures

Scientific paper

We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3).$ Special solutions of these equations, called "Maxwellians," are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted $L^{1}$-space converge to a Maxwellian in $L^{1},$ exponentially fast in time.

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